The Phragmén-Lindelöf Principle and a Class of Functional Differential Equations

Morris, G. R.; Feldstein, Alan; Bowen, Ernie W.

doi:10.1016/b978-0-12-743650-0.50048-4

Cited by 47 publications

(54 citation statements)

References 2 publications

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“…Inserting formulas (22) and (24) into Eq. (21), we obtain for A 0 (u) the asymptotic differential equation…”

Section: Perturbative Proofmentioning

confidence: 99%

“…(1) with regard to various domains of the parameters, 3 including existence and uniqueness theorems, and an extensive asymptotic analysis of the corresponding solutions have been given by Kato and McLeod [17] and Kato [16]. The investigation of such equations in the complex domain was initiated by Morris et al [22] and Oberg [23] and continued by Derfel and Iserles [6] and Marshall et al [20]. A systematic treatment of the generalized first-order pantograph equation (with matrix coefficients and also allowing for a term with rescaled derivative) is contained in the influential paper by Iserles [14], where in particular a fine geometric structure of almost-periodic solutions has been described.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On Bounded Solutions of the Balanced Generalized Pantograph Equation

Bogachev

Derfel

Molchanov

et al. 2008

The IMA Volumes in Mathematics and Its Applications

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The question about the existence and characterization of bounded solutions to linear functional-differential equations with both advanced and delayed arguments was posed in early 1970s by T. Kato in connection with the analysis of the pantograph equation, y ′ (x) = ay(qx) + by(x). In the present paper, we answer this question for the balanced generalized pantograph equation of the form −a 2 y ′′ (x)+a 1 y ′ (x)+y(x) = ∞ 0 y(αx) µ(dα), where a 1 ≥ 0, a 2 ≥ 0, a 2 1 +a 2 2 > 0, and µ is a probability measure. Namely, setting K := ∞ 0 ln α µ(dα), we prove that if K ≤ 0 then the equation does not have nontrivial (i.e., nonconstant) bounded solutions, while if K > 0 then such a solution exists. The result in the critical case, K = 0, settles a long-standing problem. The proof exploits the link with the theory of Markov processes, in that any solution of the balanced pantograph equation is an L-harmonic function relative to the generator L of a certain diffusion process with "multiplication" jumps. The paper also includes three "elementary" proofs for the simple prototype equation y ′ (x) + y(x) = 1 2 y(qx) + 1 2 y(x/q), based on perturbation, analytical, and probabilistic techniques, respectively, which may appear useful in other situations as efficient exploratory tools.

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“…Inserting formulas (22) and (24) into Eq. (21), we obtain for A 0 (u) the asymptotic differential equation…”

Section: Perturbative Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Bounded Solutions of the Balanced Generalized Pantograph Equation

Bogachev

Derfel

Molchanov

et al. 2008

The IMA Volumes in Mathematics and Its Applications

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“…In physics, chemistry, biology and engineering, a lot of problems are modelled by di erential equations, delay differential equations [10][11][12][13] and their systems [1][2][3][4][5][6][7][8][9][14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

“…These systems are di cult to solve analytically.In this paper we consider the systems of linear functional di erential equations [1][2][3][4][5][6][7][8][9] including the term y(αx + β) and advance-delay in derivatives of y . To obtain the approximate solutions of those systems, we present a matrixcollocation method by using Müntz-Legendre polynomials and the collocation points.…”

mentioning

confidence: 99%

A numerical method for solving systems of higher order linear functional differential equations

2016

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Abstract:Functional di erential equations have importance in many areas of science such as mathematical physics. These systems are di cult to solve analytically.In this paper we consider the systems of linear functional di erential equations [1][2][3][4][5][6][7][8][9] including the term y(αx + β) and advance-delay in derivatives of y . To obtain the approximate solutions of those systems, we present a matrixcollocation method by using Müntz-Legendre polynomials and the collocation points. For this purpose, to obtain the approximate solutions of those systems, we present a matrix-collocation method by using Müntz-Legendre polynomials and the collocation points. This method transform the problem into a system of linear algebraic equations. The solutions of last system determine unknown coe cients of original problem. Also, an error estimation technique is presented and the approximate solutions are improved by using it. The program of method is written in Matlab and the approximate solutions can be obtained easily. Also some examples are given to illustrate the validity of the method.

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“…We mention the paper by the second author [12] concerning the stability of the analytic solution of (1.1), two articles by the authors [3,4] about the stability of numerical solutions when 6(t) = <¡>(t) = L~xt, where L > X is an integer, and the reader is referred, for further contributions to the subject, to Carr and Dyson [5], Fox et al [9], Feldstein and Jackiewicz [8], Kato and McLeod [14], and to Morris, Feldstein, and Bowen [17] for other special cases of (1.1). Stability analysis of the exact solution of (1.1) is the theme of two forthcoming papers, Feldstein et al [7] and Iserles and Terjéki [13].…”

Section: Introductionmentioning

confidence: 99%

Stability of the Discretized Pantograph Differential Equation

Buhmann¹,

Iserles²

1993

Mathematics of Computation

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Abstract. In this paper we study discretizations of the general pantograph equation y'(t) = ay(t) + by(6(t)) + cy'((t)), f>0, y(0)=y0, where a , b , c , and yo are complex numbers and where 9 and <¡> are strictly increasing functions on the nonnegative reals with 0(0) = <^>(0) = 0 and 8(t) < t, 4>(t) < t for positive /. Our purpose is an analysis of the stability of the numerical solution with trapezoidal rule discretizations, and we will identify conditions on a , b , c and the stepsize which imply that the solution sequence {yn/^o 's DOunded or that it tends to zero algebraically, as a negative power of n . X. IntroductionIn this article we shall consider, in the most general form, the generalized pantograph equationwhere a, b, c, and yo are complex and where 8 and

(t)) is called the "neutral" term. Our principal focus in this paper is the case of proportional delays 8(t) = qt and 4>(t) = pt, where q and p are between 0 and 1. The equation (1.1) can also be considered for vector-valued y and matrices a, b, c, but we will dispense with this generalization here. There are many applications for the generalized pantograph equation. Here we only mention applications in number theory (Mahler [15]), in electrodynamics (Fox et al. [9]) and the collection of current by the pantograph of an electric locomotive (whence its name; cf. Ockendon and Tayler [18]), and in nonlinear dynamical systems (Derfel [6]). A more comprehensive list features in Iserles [12].Delay differential equations with constant delays, i.e., c = 0 and 6(t) = t-x, where one also prescribes y's values on (-t, 0), have been investigated extensively in the past (see, for instance, Bellman and Cooke [1,2] and Hale [10]).

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The Phragmén-Lindelöf Principle and a Class of Functional Differential Equations

Cited by 47 publications

References 2 publications

On Bounded Solutions of the Balanced Generalized Pantograph Equation

On Bounded Solutions of the Balanced Generalized Pantograph Equation

A numerical method for solving systems of higher order linear functional differential equations

Stability of the Discretized Pantograph Differential Equation

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